Anil K. Chopra - Earthquake Engineering for Concrete Dams

Substituting this in Eq. (2.2.1)leads to the Helmholtz equation:

(2.3.8)

The frequency response function, , for the hydrodynamic pressure when the excitation is the horizontal ground acceleration and the dam is rigid ( Figure 2.3.1a), is the solution of Eq. (2.3.8)subject to the boundary conditions of Eqs. transformed according to Eq. (2.3.7):

(2.3.9)

The frequency response function for the hydrodynamic pressure, when the excitation is the vertical ground acceleration and the dam is rigid ( Figure 2.3.1b), is the solution of Eq. (2.3.8)subject to the boundary conditions of Eqs. transformed according to Eq. (2.3.7):

(2.3.10)

The frequency response function for the hydrodynamic pressure due to horizontal acceleration of the dam vibrating in its fundamental natural vibration mode ( Figure 2.3.1c) is the solution of Eq. (2.3.8)subject to boundary conditions of Eqs. transformed according to Eq. (2.3.7):

(2.3.11)

The complex‐valued frequency response functions and are obtained using standard solution methods for boundary value problems. Specialized for the upstream face of the dam, these functions are (Fenves and Chopra 1984a)

(2.3.12a)

(2.3.12b)

(2.3.12c)

where

(2.3.13)

The frequency response functions for hydrodynamic pressure due to horizontal motions of the upstream face of the dam, given by Eqs. (2.3.12a)and (2.3.12c)are the sum of the contributions of an infinite number of natural vibration modes of the impounded water. The complex‐valued, frequency‐dependent eigenvalues μ n( ω ) satisfy Eq. (2.3.14)and the eigenfunctions ϒ n( y , ω ) are defined by Eq. (2.3.15):

(2.3.14)

(2.3.15)

If the ground motion is vertical, pressure waves do not propagate upstream resulting in the much simpler frequency response function ( Eq. (2.3.12b)), which is independent of the x ‐coordinate.

Non‐absorptive Reservoir BottomFor a rigid, non‐absorptive reservoir bottom, as mentioned earlier, ξ = 0 and α = 1; the eigenvalues μ n( ω ) and eigenfunctions ϒ n( y , ω ) are real‐valued and independent of the excitation frequency:

(2.3.16)

where are the natural vibration frequencies of the impounded water with rigid non‐absorptive reservoir bottom, and

(2.3.17)

Then Eqs. (2.3.12a)and (2.3.12b)specialize to

(2.3.18)

(2.3.19)

These are the same as the equations defining hydrodynamic pressures on the upstream face of a rigid dam for a non‐absorptive reservoir bottom presented in Chopra (1967).

Incompressible WaterNeglecting compressibility of water is equivalent to assuming the speed C of the hydrodynamic pressure waves to be infinite. The limits of Eqs. (2.3.18)and (2.3.19)as C → ∞ result in

(2.3.20)

(2.3.21)

Observe that the hydrodynamic pressure functions and are now independent of the excitation frequency, and is equal to the hydrostatic pressure.

The complex‐valued frequency response functions for the hydrodynamic force on a rigid dam due to horizontal and vertical ground acceleration are computed from Eqs. (2.3.12a)and (2.3.12b)as the integral of over the depth of water. The real and imaginary components as well as the absolute value of , normalized with respect to the hydrostatic force F st= ρ g H 2/2, are plotted in Figures 2.3.2and 2.3.3for five different values of α as a function of the dimensionless excitation frequency , where ( Eq. (2.3.16)) is the first natural vibration frequency of the impounded water with non‐absorptive reservoir bottom. When presented in this non‐dimensional form, the plots apply to fluid domains of any depth. The real and imaginary components represent the in‐phase (or 180°‐out‐of‐phase) and 90°‐out‐of‐phase hydrodynamic forces relative to the harmonic ground acceleration.